I noticed something weird about traffic lights that have a separate red/green arrow for people turning left. Sometimes the two opposite left turns go at the same time, and other times, the left turns each go when people starting out the same direction have a green light. (Hopefully I phrased that clearly enough.) Why is this?
In a lot of places, there are sensors buried in the left turn lanes that tell the traffic light controller whether there are any cars waiting to make the left turn.
So, let’s consider an intersection where there’s a left turn lane for the folks heading east, and one for the folks heading west. If there are cars waiting in both of those lanes, the controller will give each of them the green arrow, while maintaining red for the folks going straight.
However, if there are eastbound cars waiting to make a left, but no westbound cars doing so, the controller will give the eastbound folks a green arrow and a green light for going straight, while maintaining red for the westbound folks. No sense making the people heading straight in the eastbound lanes wait for westbound left-turners who aren’t there.
BTW, this is also why, sometimes, you don’t get a green arrow to make your left turn - you’ve pulled forward too far, so you’re not over the buried sensor, so the traffic light controller doesn’t know you’re waiting to make a left.
I have a feeling that time of day matters too - in periods when traffic flow in one direction or another is recognisably different (say, going into the city in the morning, leaving the city at night) the regularity of changing, and the rules applied to them, are different.
But I’m just going by observation.
I think you mean that sometimes both north and southbound (or east and west) left turns go and sometimes all southbound traffic is held up while all northbound traffic, including left turners, goes.
I think this is done on the basis of traffic surveys. The programming of modern traffic signals can be quite sophisticated.
Suppose we have a large factory where employees go to work at 8:00 AM and leave work at 5:00 PM. It can assumed that north of the factory we need to give southbound traffic more time with green than northbound in the morning and less in the afternoon. So, in order to keep southbound traffic moving smoothly in the morning we give both straight through and turning southbound drivers first crack and give them a longer time, in order to clear the intersection. Then in the afternoon, north of the factory, the timing would be reversed. During hours when people aren’t going to work the times would be equal. The streets south, east and west would be handled the same way.
Just my WAG
You might be right, especially since TJdude825 hails from CA. In ole Virginny, alas, nothing about the highway department is nearly that smart! 
At the intersection you may see a metal cabinet standing behind one of the signal poles. This box has the controller in it. This computer can be programmed for any scenario by the traffice engineer, such as the factory example above. Usually the side street turn arrows & green change according to wait times i.e. if car is detected on side street & traffic is detected on main street then wait 30 sec before changing signal.
The traffic engineer can also tweak the settings during times such as mega-holiday-shopper-blitz-weekend at the mall.
Some signals are looped together and work in concert. Have you ever caught a green light for 5 or 6 signals in a row? That’s when they are synchronized.
Now they even have controllers which are connected to the computer in the traffic engineer’s office. When used with video cameras at the intersection, the engineer can program the signal in real time to help with traffic flow.
An aside: next time you are sitting at the light look up to the signal mast. You may see a small transmitter mounted on the mast. This is the Opticon, which is tuned to see a transmitter mounted in the light bar of fire trucks, ambulances, & police cars. The Opticon, when approached by such an emergency signal, turns the lights red all the way 'round the intersection. After the emergency vehicle leaves the intersection, the lights will flash red & yellow for a few seconds while the controller re-sets itself.
Sometimes. Sometimes they use “smart lights” like Early Out mentioned. It depends. (Note that just because there isn’t a sawcut in the road, that doesn’t mean there isn’t a sensor–don’t pull up past the stop line unless there are cars behind you to trip the sensor.) Sometimes they are based on a pattern, derived from traffic/engineering data, but different from what David Simmons described. For example, if the road width is such that left-turning cars may prevent those going straight from going through, then they may get the left light first. Sometimes the data shows a high percentage of left turns going one way, but a low percentage in the opposite direction, then they may find a pattern that seems goofy on the face of it, but actually helps to move more cars.
If you are curious about current traffic research, you can check out some papers at the Traffic Forum. Check out the links page, there are a bunch of Java Applets modeling traffic.